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Contents

Calculation

In principle, ρ is simply a special case of the Pearson product-moment coefficient in which two sets of data and are converted to rankings and before calculating the coefficient.[1] In practice, however, a simpler procedure is normally used to calculate ρ. The raw scores are converted to ranks, and the differences between the ranks of each observation on the two variables are calculated.

The first step is to sort this data by the second column. Next, two more columns are created ( and ). The last of these columns () is assigned 1,2,3,...n, and then the data is sorted by the first original column (). The first of the newly created columns () is assigned 1,2,3,...n. Then a column is created to hold the differences between the two rank columns ( and ). Finally another column should be created. This is just column squared.

After doing this process with the example data you should end up with something like:

The values in the column can now be added to find . The value of n is 10. So these values can now be substituted back into the equation,

which evaluates to which shows that the correlation between IQ and hours spent watching TV is very low (barely any correlation). In the case of ties in the original values, this formula should not be used. Instead, the Pearson correlation coefficient should be calculated on the ranks (where ties are given ranks, as described above).

Determining significance

The modern approach to testing whether an observed value of ρ is significantly different from zero (we will always have 1 ≥ ρ ≥ −1) is to calculate the probability that it would be greater than or equal to the observed ρ, given the null hypothesis, by using a permutation test. This approach is almost always superior to traditional methods, unless the data set is so large that computing power is not sufficient to generate permutations, or unless an algorithm for creating permutations that are logical under the null hypothesis is difficult to devise for the particular case (but usually these algorithms are straightforward).

Although the permutation test is often trivial to perform for anyone with computing resources and programming experience, traditional methods for determining significance are still widely used. The most basic approach is to compare the observed ρ with published tables for various levels of significance. This is a simple solution if the significance only needs to be known within a certain range or less than a certain value, as long as tables are available that specify the desired ranges. A reference to such a table is given below. However, generating these tables is computationally intensive and complicated mathematical tricks have been used over the years to generate tables for larger and larger sample sizes, so it is not practical for most people to extend existing tables.

An alternative approach available for sufficiently large sample sizes is an approximation to the Student's t-distribution with degrees of freedom N-2. For sample sizes above about 20, the variable

has a Student's t-distribution in the null case (zero correlation). In the non-null case (i.e. to test whether an observed ρ is significantly different from a theoretical value, or whether two observed ρs differ significantly) tests are much less powerful, though the t-distribution can again be used.

A generalization of the Spearman coefficient is useful in the situation where there are three or more conditions, a number of subjects are all observed in each of them, and we predict that the observations will have a particular order. For example, a number of subjects might each be given three trials at the same task, and we predict that performance will improve from trial to trial. A test of the significance of the trend between conditions in this situation was developed by E. B. Page and is usually referred to as Page's trend test for ordered alternatives.

External links

Spearman's rank correlation: Simple notes for students with an example of usage by biologists and a spreadsheet for Microsoft Excel for calculating it (a part of materials for a Research Methods in Biology course).